# How do I write #(5/(1*2))+(5/(2*3))+(5/(3*4))+...+(5/n(n+1))+...#in summation notation, and how can I tell if the series converges?

The given series:

Hence, the given series is converging

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Please see some comlementary details below.

I added a few more details

The partial fraction decomposition is

Compare the numerators

Therefore,

Therefore,

And

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The series can be written in summation notation as:

[ \sum_{n=1}^{\infty} \frac{5}{n(n+1)} ]

To determine if the series converges, you can use the Ratio Test. Let (a_n = \frac{5}{n(n+1)}). Then:

[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{\frac{5}{(n+1)(n+2)}}{\frac{5}{n(n+1)}} \right| = \lim_{n \to \infty} \left| \frac{n}{n+2} \right| = 1 ]

Since the limit is equal to 1, the Ratio Test is inconclusive. You can try another convergence test, such as the Integral Test or Comparison Test, to determine the convergence of the series.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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